Induced permittivity increment of electrorheological fluids in an applied electric field in association with chain formation: A Brownian dynamics simulation study

We report Brownian dynamics simulation results for the relative permittivity of electrorheological (ER) fluids in an applied electric field. The relative permittivity of an ER fluid can be calculated from the Clausius-Mosotti (CM) equation in the small applied field limit. When a strong field is applied, however, the ER spheres are organized into chains and assemblies of chains in which case the ER spheres are polarized not only by the external field but by each other. This manifests itself in an enhanced dielectric response, e.g., in an increase in the relative permittivity. The correction to the relative permittivity and the time dependence of this correction is simulated on the basis of a model in which the ER particles are represented as polarizable spheres. In this model, the spheres are also polarized by each other in addition to the applied field. Our results are qualitatively similar to those obtained by Horváth and Szalai experimentally [Phys. Rev. E 86, 061403 (2012)]. We report characteristic time constants obtained from biexponential fits that can be associated with the formation of pairs and short chains as well as with the aggregation of chains. The electric field dependence of the induced dielectric increment reveals the same qualitative behavior that experiments did: three regions with different slopes corresponding to different aggregation processes are identified.

Figure 1

Time dependence of $\mathrm{\Delta }\epsilon$ for ${\alpha }^{*}=0.03$. The symbols with error bars are simulated data. The error bars have been computed from the variance of the data in the consecutive and independent periods. The lines are biexponential fits [Eq. (3)]. The numbers near the curves indicate the values of the reduced electric field, $E^{*}$. The inset shows the experimental data of Horváth and Szalai [9] for comparison. The numbers in the inset indicate electric field strengths in MV/m.

Figure 2

The characteristic times ${\tau }_1$ and ${\tau }_2$ (the latter is larger by an order of magnitude) as functions of (a) $E^{*}$ and (b) ${\alpha }^{*}{E}^{*}$ for various values of ${\alpha }^{*}$. The inset shows the experimental data of Horváth and Szalai [9] (from their Table 1) for the two different values of the viscosity they considered (black and red refer to $\eta =0.34$ and $0.97\mathrm{Pa}\mathrm{s}$, respectively). The error bars of ${\tau }_1$ and ${\tau }_2$ estimated from the Levenberg-Marquardt algorithm [41] are within the size of the symbols.

Figure 3

The equilibrium ($t\to \infty$) values of the one-particle dipolar energy, ${\left(u^{\mathrm{dip}}\right)}^{*}$ (top-left panel), the diffusion constant, $D^{*}$ (top-right panel), the induced permittivity increment, $\mathrm{\Delta }\epsilon$ (bottom-left panel), and the average chain length, $s_{\mathrm{av}}$ (bottom-right panel), for various values of ${\alpha }^{*}$. The vertical dashed lines indicate the $E_{\mathrm{h}}^{*}$ field strengths at which the regions with different slopes meet (see Fig. 4 for more explanation). The error bars are those computed at $t^{*}=4500$ from the variance of the values in the periods. The error bars for ${\left(u^{\mathrm{dip}}\right)}^{*}$ and $\mathrm{\Delta }\epsilon$ are within the size of the symbols.

Figure 4

The equilibrium ($t\to \infty$) value of $\mathrm{\Delta }\epsilon$ as a function of $E^{*}$ for ${\alpha }^{*}=0.03$. The inset shows the experimental data from Fig. 4 of Horváth and Szalai [9]. The inset of the inset shows the results for small $E^{*}$. The solid blue lines indicate the slopes, while dashed blue lines indicate the crosses of the solid lines defining the threshold field strengths $E_1$ and $E_{\mathrm{h}}$.

Figure 5

Equilibrium limits of the chain length distribution, $n_s$ (left), and radial distribution functions, $g\left(r^{*}\right)$ (right), for ${\alpha }^{*}=0.03$. The curves are obtained by averaging over 10 blocks at the end of $M_E$ simulation periods and averaging over periods. Snapshots are shown in the middle in front and top view. The electric field strength increases from top to bottom.

Figure 6

Snapshots for a very large electric field, $E^{*}=149.07$, for ${\alpha }^{*}=0.03$.